For $A, B \subseteq \omega$ we set $A \leq_{\text{inj}} B$ if there is an injective and order-preserving map $f:\omega\to \omega$ , such that $f(A)$ is a down-set of $B$. It is easy to see that $\leq_{\text{inj}}$ is an ordering relation on ${\cal P}(\omega)$.

Is $({\cal P}(\omega), \leq_{\text{inj}})$ a lattice? Is it distributive?